I have been reading the stackexchange questions on enhanced symmetries in string theory, the Leech lattice, monstrous moonshine, etc. , and I have a question to ask.

An astute commentator pointed out with enhanced symmetries, massless vector bosons have to couple in the Yang-Mills fashion. Only if the massless vectors come in a Lie algebra will that be possible.

Imagine bosonic string theory. Compactify 24 spatial dimensions over the Leech lattice or odd Leech lattice. With respect to the uncompactified dimensions (one spatial and one timelike), we get 196560 massless vector bosons, but they can't be part of any Lie algebra. But that's OK because massless vector bosons in 2 dimensions have zero norm.

Let's try again. Compactify 23 spatial dimensions over the shorter Leech lattice. There are 2 spatial and one timelike uncompactified dimensions. Over this space, we have 4600 massless vector bosons, but they're not part of any Lie algebra either. But that's also OK because massless vector bosons in 3 dimensions are physically indistinguishable from massless scalar bosons. The reason for this is there is no helicity.

If we try to find a similar example over at least 4 uncompactified dimensions, where massless vector bosons exist for certain and aren't equivalent to anything else, we discover all unimodular lattices in 22 dimensions or less are part of a root system. Thus, we always have a Lie algebra.

What is the magical mechanism in string theory enforcing this sort of self-consistency? Is string theory only self-consistent because of accidental coincidences?

2 Answers
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I think you're confused about the structure of the Leech lattice. The minimum length squared is 4, not 2 as for the root lattices of simply laced Lie algebras. The mass of a state with momentum $p \in \Lambda_{Leech}$ and no oscillator excitations is $p^2/2-1$, so the 196560 states you are talking about are massive, not massless. Before doing an asymmetrical $Z_2$ orbifold projection to get the monster theory there are $24$ massless states, there are just the gauge bosons of $U(1)^{24}$. The $Z_2$ orbifold removes these states and leaves you with a theory which just has massive states. Similar comments apply to your other examples. By the way, although it is not directly relevant to your question, there is an algebraic structure one can define involving the vertex operators for the $196884$ states at the first mass level. It is known as the Griess algebra and its automorphism group is the Monster group.

It looks miraculous. However, one may also adopt a less religious attitude to those "coincidences".

Well, they're not coincidences at all. They're provable mathematical facts. One can explicitly show that the relevant consistency conditions - such as the absence of ghosts - are satisfied by bosonic string theory compactified on any torus of dimension 22 or lower. These no-ghost theorems are even parts of some textbooks.

(Other, more special consistency conditions, such as stability of the vacuum, clearly fail in bosonic string theory.)

Also, it can be showed, using the spacetime perspective - effective field theory - that in 3+1 large dimensions or higher (as you correctly clarified), charged spin-one bosons may only consistently interact if they're a part of Yang-Mills theory.

So in this way, combining the two totally mathematical steps above, one can prove - and indeed, string theorists have proved - that all unimodular lattices in 22 or less dimensions are a part of a root system. (Jeff Harvey, in the other answer, strengthens the statement of yours: you wouldn't get a contradiction even for 23- or 24-dimensional lattices because the shortest nonzero vectors in the lattice already correspond to massive particles. But I will only talk about your more modest statement that assumes that the dimension of the lattice is 22 or smaller. I would also have a problem with your condition of "unimodularity". You really want to study lattices with many $p^2=2$ points and no other condition or, if you want to study both left-movers and right-movers, the discussion is about even self-dual lattices in $d_{22}+d_{22}$ dimensions. But let me focus on the philosophical content here, not the technicalities.)

One could transform this proof into mathematical jargon that will hide all the traces of its stringy origin: but it's still true that string theory is a natural engine that produces remarkable proofs of many amazing facts in mathematics. It tells you how you should optimally proceed if you want to prove an important theorem in mathematics. So what you call is not an "accidental coincidence" - it's a deep, not-at-all coincidental sharp mathematical theorem, and it can be proved by stringy physics-inspired methods.

So this string theory approach also proves a metaphysical statement - namely that this observation is pretty important. It is not an irrelevant coincidence. It has a deeper meaning. Your opinion that it is just a "coincidence" has been falsified. In the same way, your expectation that this "coincidence" may only be proved by checking the lattice one by one has also been proved incorrect. It's a qualitative statement that has a qualitative, conceptual proof - even though this fact may have failed to be self-evident at the beginning.

It's not necessarily the case that mathematicians must first think of all possible structures, all possible theorems, and all possible proofs, and these results are then used by physicists to describe the physical world. When physicists are working on a theory that is as deep as string theory, the chronology is often the opposite one: they're finding structures, theorems, and/or their proofs that are unknown to mathematicians of their time.

Needless to say, such things may occur with physics models that are easier than string theory, too. Witten has used Chern-Simons theory to prove various things about knots and other things. In string theory, those results are omnipresent. String theory produces lots of such results that would look like "accidental coincidences" i.e. "miracles" to a person who doesn't understand any string theory.

Let me mention one not-too-quoted but amazing example. There is an identity between sums of products of Bernoulli numbers that was unknown before 2005:

Look at equation (B.5). It is similar to the Euler-Ramanujan identities. However, it's true and Martin Schnabl discovered this identity - that can be proved (or, at least, certainly checked) once we know it's true while he was solving open string field theory for the disappeared bosonic D25-brane. So this seemingly "miraculous" or "accidentally coincidental" identity follows from the fact that "the space where the D-branes has totally disappeared solves the equations of motion". By reparameterizations that only look natural with the whole string-theory cannon, one may reduce these identities to some much more obvious, but more formal, identities.